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Linear Algebra LINEAR ALGEBRA by Alexander Givental s u m ι Σ τ σ δ α Sumizdat s u m Published by Sumizdat ι Σ τ 5426 Hillside Avenue, El Cerrito, California 94530, USA σ δ α http://www.sumizdat.org University of California, Berkeley Cataloging-in-Publication Data Givental, Alexander Linear Algebra / by Alexander Givental ; El Cerrito, Calif. : Sumizdat, 2009. iv, 200 p. : ill. ; 23 cm. Includes bibliographical references and index. ISBN 978-0-9779852-4-1 1. Linear algebra. I. Givental, Alexander. Library of Congress Control Number: c 2009 by Alexander Givental All rights reserved. Copies or derivative products of the whole work or any part of it may not be produced without the written permission from Alexander Givental ([email protected]), except for brief excerpts in connection with reviews or scholarly analysis. Credits Editing: Alisa Givental. Art advising: Irina Mukhacheva, http://irinartstudio.com Cataloging-in-publication: Catherine Moreno, Technical Services Department, Library, UC Berkeley. A Layout, typesetting and graphics: using L TEX and Xfig. Printing and binding: Thomson-Shore, Inc., http://www.tshore.com Member of the Green Press Initiative. 7300 West Joy Road, Dexter, Michigan 48130-9701, USA. Offset printing on 30% recycled paper; cover: by 4-color process on Kivar-7. ISBN 978-0-9779852-4-1 Contents 1 A Crash Course 1 1 Vectors .......................... 1 2 QuadraticCurves..................... 9 3 ComplexNumbers .................... 17 4 ProblemsofLinearAlgebra . 23 2 Dramatis Personae 31 1 Matrices.......................... 31 2 Determinants ....................... 41 3 VectorSpaces....................... 59 3 Simple Problems 71 1 DimensionandRank................... 71 2 GaussianElimination. 83 3 TheInertiaTheorem................... 97 4 Eigenvalues 117 1 TheSpectralTheorem . .117 2 JordanCanonicalForms . 137 Hints ..............................149 Answers .............................153 Bibliography .........................157 Index ..............................158 iii iv Foreword “Mathematics is a form of culture. Mathematical ignorance is a form of national culture.” “In antiquity, they knew few ways to learn, but modern pedagogy discovered 1001 ways to fail.” August Dumbel1 1From his eye-opening address to NCVM, the National Council of Visionaries of Mathematics. v vi Chapter 1 A Crash Course 1 Vectors Operations and their properties The following definition of vectors can be found in elementary geom- etry textbooks, see for instance [4]. A directed segment −−→AB on the plane or in space is specified by an ordered pair of points: the tail A and the head B. Two directed segments −−→AB and −−→CD are said to represent the same vector if they are obtained from one another by translation. In other words, the lines AB and CD must be parallel, the lengths AB and CD must be equal, and the segments must point toward the| same| of| the| two possible directions (Figure 1). B B v D u A C u+v C A Figure 1 Figure 2 A trip from A to B followed by a trip from B to C results in a trip from A to C. This observation motivates the definition of the 1 2 Chapter 1. A CRASH COURSE vector sum w = v + u of two vectors v and u: if −−→AB represents v and −−→BC represents u then −→AC represents their sum w (Figure 2). The vector 3v = v + v + v has the same direction as v but is 3 times longer. Generalizing this example one arrives at the definition of the multiplication of a vector by a scalar: given a vector v and a real number α, the result of their multiplication is a vector, denoted αv, which has the same direction as v but is α times longer. The last phrase calls for comments since it is literally true only for α> 1. If 0 <α< 1, being “α times longer” actually means “shorter.” If α < 0, the direction of αv is in fact opposite to the direction of v. Finally, 0v = 0 is the zero vector represented by directed segments −→AA of zero length. Combining the operations of vector addition and multiplication by scalars we can form expressions αu+βv+...+γw. They are called linear combinations of the vectors u, v, ..., w with the coefficients α, β, ..., γ. The pictures of a parallelogram and parallelepiped (Figures 3 and 4) prove that the addition of vectors is commutative and associa- tive: for all vectors u, v, w, u + v = v + u and (u + v)+ w = u + (v + w). u u+v+w w v+u v+w v v u+v u+v v u u Figure 3 Figure 4 From properties of proportional segments and similar triangles, the reader will easily derive the following two distributive laws: for all vectors u, v and scalars α, β, (α + β)u = αu + βv and α(u + v)= αu + αv. 1. Vectors 3 Coordinates From a point O in space, draw three directed segments −→OA, −−→OB, and −−→OC not lying in the same plane and denote by i, j, and k the vectors they represent. Then every vector u = −−→OU can be uniquely written as a linear combination of i, j, k (Figure 5): u = αi + βj + γk. The coefficients form the array (α, β, γ) of coordinates of the vec- tor u (and of the point U) with respect to the basis i, j, k (or the coordinate system OABC). αi βj U γk u C B OA Figure 5 Multiplying u by a scalar λ or adding another vector u′ = α′i + β′j + γ′k, and using the above algebraic properties of the operations with vectors, we find: λu = λαi+λβj+λγk, and u+u′ = (α+α′)i+(β +β′)j+(γ +γ′)k. Thus, the geometric operations with vectors are expressed by com- ponentwise operations with the arrays of their coordinates: λ(α, β, γ) = (λα, λβ, λγ), (α, β, γ) + (α′, β′, γ′) = (α + α′, β + β′, γ + γ′). What is a vector? No doubt, the idea of vectors is not new to the reader. However, some subtleties of the above introduction do not easily meet the eye, and we would like to say here a few words about them. 4 Chapter 1. A CRASH COURSE As many other mathematical notions, vectors come from physics, where they represent quantities, such as velocities and forces, which are characterized by their magnitude and direction. Yet, the popular slogan “Vectors are magnitude and direction” does not qualify for a mathematical definition of vectors, e.g. because it does not tell us how to operate with them. The computer science definition of vectors as arrays of numbers, to be added “apples with apples, oranges with oranges,” will meet the following objection by physicists. When a coordinate system rotates, the coordinates of the same force or velocity will change, but the numbers of apples and oranges won’t. Thus forces and velocities are not arrays of numbers. The geometric notion of a directed segment resolves this problem. Note however, that calling directed segments vectors would constitute abuse of terminology. Indeed, strictly speaking, directed segments can be added only when the head of one of them coincides with the tail of the other. So, what is a vector? In our formulations, we actually avoided answering this question directly, and said instead that two directed segments represent the same vector if . Such wording is due to ped- agogical wisdom of the authors of elementary geometry textbooks, because a direct answer sounds quite abstract: A vector is the class of all directed segments obtained from each other by translation in space. Such a class is shown in Figure 6. Figure 6 This picture has another interpretation: For every point in space (the tail of an arrow), it indicates a new position (the head). The ge- ometric transformation in space defined this way is translation. This leads to another attractive point of view: a vector is a translation. Then the sum of two vectors is the composition of the translations. 1. Vectors 5 The dot product This operation encodes metric concepts of elementary Euclidean ge- ometry, such as lengths and angles. Given two vectors u and v of lengths u and v and making the angle θ to each other, their dot product| (also| called| | inner product or scalar product) is a number defined by the formula: u, v = u v cos θ. h i | | | | Of the following properties, the first three are easy (check them!): (a) u, v = v, u (symmetricity); h i h i (b) u, u = u 2 > 0 unless u = 0 (positivity); h i | | (c) λu, v = λ u, v = u, λv (homogeneity); h i h i h i (d) u + v, w = u, w + v, w (additivity with respect to the first factor).h i h i h i To prove the last property, note that due to homogeneity, it suf- fices to check it assuming that w is a unit vector, i.e. w =1. In | | this case, consider (Figure 7) a triangle ABC such that −−→AB = u, −−→BC = v, and therefore −→AC = u + v, and let −−→OW = w. We can consider the line OW as the number line, with the points O and W representing the numbers 0 and 1 respectively, and denote by α, β, γ the numbers representing perpendicular projections to this line of the vertices A,B,C of the triangle. Then −−→AB, w = β α, −−→BC, w = γ β, and −→AC, w = γ α. h i − h i − h i − The required identity follows, because γ α = (γ β) + (β α). − − − B u v C A u+v OW α w β γ Figure 7 Combining the properties (c) and (d) with (a), we obtain the following identities, expressing bilinearity of the dot product (i.e. 6 Chapter 1. A CRASH COURSE linearity with respect to each factor): αu + βv, w = α u, w + β v, w h i h i h i w, αu + βv = α w, u + β w, v .
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